Abstract

In 1962, Miyamoto and Wolf succeeded in formulating the boundary-diffraction-wave theory for a general incident wave. In this paper, the theory is applied to the aperture having an arbitrary transmittance distribution, and it is found that every point where the gradient of the transmittance distribution is not zero is the origin of a secondary wave. The boundary diffraction wave can be expressed by the wave that originates at the points where the gradient of the transmittance distribution is the Dirac delta function. Hence, it seems reasonable that the diffraction wave, which is generated from the aperture point where the gradient of transmittance is not zero, is more general for discussion of diffraction problems.

© 1971 Optical Society of America

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References

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  1. T. Young, Phil. Trans. Roy. Soc. (London) 20, 26 (1802).
  2. G. A. Maggi, Ann. Mat. (Rome) IIa 16, 21 (1888).
    [Crossref]
  3. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed (Oxford U. P., New York, 1950) Sec 2.
  4. A. Rubinowicz, Ann. Physik 53, 257 (1917).
    [Crossref]
  5. A. Rubinowicz, Phys. Rev. 54, 931 (1938).
    [Crossref]
  6. A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).
  7. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer, Berlin, 1967).
  8. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec.8.9.
  9. A. Sommerfeld, Optics (Academic, New York, 1954), p. 262.
  10. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962)
    [Crossref]
  11. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).
    [Crossref]
  12. A. Rubinowicz, in Progress in Optics, IV, edited by E. Wolf (North-Holland, Amsterdam, 1965), p. 201.
  13. In the special case in which a wave incident upon an aperture is plane or spherical, U(G)(P) represents a wave propagated in accordance with the laws of geometrical optics (see Ref. 11).
  14. A. Rubinowicz, J. Opt. Soc. Am. 52, 717 (1962); also Ref. 12
    [Crossref]
  15. See Ref. 10, p. 621.
  16. For mathematical convenience, the transmittance on the edge of an aperture was changed from that discussed in Sec. I to the transmittance shown in Figs. 4 or 5. Hence, for the one-dimensional case, the transmittance on the edge Γ may be defined as limy→+y1T in the first term of Eq. (14).
  17. The comment,“A boundary diffraction wave may be thought of as arising from the scattering of the incident radiation by the boundary of the aperture,” is found in Ref. 8,p. 452.

1962 (3)

1953 (1)

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

1938 (1)

A. Rubinowicz, Phys. Rev. 54, 931 (1938).
[Crossref]

1917 (1)

A. Rubinowicz, Ann. Physik 53, 257 (1917).
[Crossref]

1888 (1)

G. A. Maggi, Ann. Mat. (Rome) IIa 16, 21 (1888).
[Crossref]

1802 (1)

T. Young, Phil. Trans. Roy. Soc. (London) 20, 26 (1802).

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed (Oxford U. P., New York, 1950) Sec 2.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec.8.9.

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed (Oxford U. P., New York, 1950) Sec 2.

Maggi, G. A.

G. A. Maggi, Ann. Mat. (Rome) IIa 16, 21 (1888).
[Crossref]

Miyamoto, K.

Rubinowicz, A.

A. Rubinowicz, J. Opt. Soc. Am. 52, 717 (1962); also Ref. 12
[Crossref]

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

A. Rubinowicz, Phys. Rev. 54, 931 (1938).
[Crossref]

A. Rubinowicz, Ann. Physik 53, 257 (1917).
[Crossref]

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer, Berlin, 1967).

A. Rubinowicz, in Progress in Optics, IV, edited by E. Wolf (North-Holland, Amsterdam, 1965), p. 201.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 262.

Wolf, E.

Young, T.

T. Young, Phil. Trans. Roy. Soc. (London) 20, 26 (1802).

Acta Phys. Polon. (1)

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

Ann. Mat. (Rome) IIa (1)

G. A. Maggi, Ann. Mat. (Rome) IIa 16, 21 (1888).
[Crossref]

Ann. Physik (1)

A. Rubinowicz, Ann. Physik 53, 257 (1917).
[Crossref]

J. Opt. Soc. Am. (3)

Phil. Trans. Roy. Soc. (London) (1)

T. Young, Phil. Trans. Roy. Soc. (London) 20, 26 (1802).

Phys. Rev. (1)

A. Rubinowicz, Phys. Rev. 54, 931 (1938).
[Crossref]

Other (9)

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed (Oxford U. P., New York, 1950) Sec 2.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Springer, Berlin, 1967).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec.8.9.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 262.

See Ref. 10, p. 621.

For mathematical convenience, the transmittance on the edge of an aperture was changed from that discussed in Sec. I to the transmittance shown in Figs. 4 or 5. Hence, for the one-dimensional case, the transmittance on the edge Γ may be defined as limy→+y1T in the first term of Eq. (14).

The comment,“A boundary diffraction wave may be thought of as arising from the scattering of the incident radiation by the boundary of the aperture,” is found in Ref. 8,p. 452.

A. Rubinowicz, in Progress in Optics, IV, edited by E. Wolf (North-Holland, Amsterdam, 1965), p. 201.

In the special case in which a wave incident upon an aperture is plane or spherical, U(G)(P) represents a wave propagated in accordance with the laws of geometrical optics (see Ref. 11).

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Figures (7)

F. 1
F. 1

Notation used in this paper.

F. 2
F. 2

Shape of aperture edge.

F. 3
F. 3

Example of partial edge:γ1.

F. 4
F. 4

Cross section of one-dimensional transmittance distribution across aperture edge.

F. 5
F. 5

Continuous component of transmittance distribution:Tc.

F. 6
F. 6

Discontinuous component of transmittance distribution:Td.

F. 7
F. 7

Transmittance distribution on tangent parallel to the x axis.

Equations (43)

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V ( P , t ) = U ( P ) exp ( i ω t ) .
U ( K ) ( P ) = A V ( Q , P ) · n d S ,
V ( Q , P ) = 1 4 π [ U ( Q ) grad Q ( exp ( i k s ) s ) ( exp ( i k s ) s ) grad Q U ( Q ) ] .
div Q V ( Q , P ) = 0 .
V ( Q , P ) = curl Q W ( Q , P ) ,
U ( K ) ( P ) = U ( G ) ( P ) + U ( B ) ( P ) ,
U ( G ) ( P ) = A lim σ j 0 Γ j W · Q d l
U ( B ) ( P ) = Γ W · l d l .
W ( Q , P ) = exp ( i k s ) 4 π s ŝ × 0 exp ( i k μ ) grad U ( r + μ ŝ ) d μ + W
1 4 π ŝ 0 exp ( i k μ ) grad U ( r + μ ŝ ) d μ .
V 0 ( Q , P ) = 1 4 π [ U 0 ( Q ) grad Q ( exp ( i k s ) s ) ( exp ( i k s ) s ) grad Q U 0 ( Q ) ] .
V ( Q , P ) = T ( Q ) V 0 ( Q , P ) .
U ( K ) ( P ) = A T ( Q ) V 0 ( Q , P ) · n d S = A T ( Q ) curl W 0 ( Q , P ) · n d S .
T curl W 0 = curl ( T W 0 ) ( grad T ) × W 0 .
U ( K ) ( P ) = A curl ( T W 0 ) · n d S A { ( grad T ) × W 0 } · n d S .
U ( K ) ( P ) = A F j , T ( P ) + Γ T W 0 · l d l A { ( grad T ) × W 0 } · n d S .
( grad T ) × W 0 · n d S = ( grad T ) i rreg × W 0 · n d S ( grad T ) reg × W 0 · n d S ,
y γ 1 = f γ 1 ( x ) for x A < x < x B
x γ 1 = g γ 1 ( y ) for y A < y < y B
γ 1 T W 0 · l d l = γ 1 T W 0 · i d x + γ 1 T W 0 · j d y
= γ 1 T W 0 x d x + γ 1 T W 0 y d y ,
γ 1 T W 0 x d x = T W 0 x δ ( y y γ 1 ) d x d y = ( T W 0 x ) y = y γ 1 δ ( y y γ 1 ) d x d y
γ 1 T W 0 y d y = Σ T W 0 y δ ( x x γ 1 ) d x d y = Σ ( T W 0 y ) x = x γ 1 δ ( x x γ 1 ) d x d y ,
( T y ) y = y γ 1 = T d y = T y γ 1 δ ( y y γ 1 ) .
γ 1 T W 0 x d x = Σ ( T y ) y = y γ 1 W 0 x d x d y = Σ { ( T y ) W 0 x } y = y γ 1 d x d y .
γ 1 T W 0 y d y = Σ ( T x ) x = x γ 1 W 0 y d x d y = Σ { ( T x ) W 0 y } x = x γ 1 d x d y ,
γ 1 T W 0 · l d l = Σ [ { ( T y ) W 0 x } y = y γ 1 { ( T x ) W 0 y } x = x γ 1 ] d x d y ,
γ 1 T W 0 · l d l = Σ { ( grad T ) × W 0 } γ 1 · n d x d y .
Y ( x ) = 1 π [ tan 1 ( x ) + π 2 ] ,
P ( x ) = exp ( x 2 / 4 ) ,
P ( x ) = 2 ( π ) 1 2 x 4 π 3 2 exp ( x 2 4 ) = 2 ( π ) 1 2 · δ ( x ) .
( T x ) W 0 y d x d y = lim 0 T ( x A ) P ( x x A ) W 0 y d x d y = lim 0 2 ( π ) 1 2 T ( x A ) ( W 0 y x ) x = x A d y .
( T x ) W 0 y d x d y = 0 .
U ( K ) ( P ) = F j · T ( P ) { ( grad ) × W 0 } · n d S ,
V ( Q , P ) = 1 4 π [ U ( Q ) grad Q ( exp ( i k s ) s ) ( exp ( i k s ) s ) grad Q U ( Q ) ] .
U ( Q ) = T ( Q ) U 0 ( Q ) ,
{ grad Q ( T U 0 ) } · n = T ( grad Q U 0 ) · n + U 0 ( grad Q T ) · n .
grad Q T ( Q ) = T x i + T y j ,
n = k ,
( grad Q T ) · n = ( T x ) ( i · k ) + ( T y ) ( j · k ) = 0 ,
{ grad Q ( T U 0 ) } · n = T ( grad Q U 0 ) · n .
V 0 ( Q , P ) = 1 4 π [ U 0 ( Q ) grad Q ( exp ( i k s ) s ) ( exp ( i k s ) s ) grad Q U 0 ( Q ) ] .
V ( Q , P ) · n = 1 4 π [ T ( Q ) U 0 ( Q ) grad Q ( exp ( i k s ) s ) ( exp ( i k s ) s ) T ( Q ) grad Q U 0 ( Q ) ] · n = T ( Q ) V 0 ( Q , P ) · n .